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SDP hierarchies and quantum states Aram Harrow (MIT) Simons Institute 2014.1.17 a theorem Let M2R+m£n. Say that a set S⊆[n]k is δ-good if ∃φ:[m]k S such that ∀(j1, …, jk)∈S, f(k,δ):= max{ |S| : ∃S⊆[n]k, S is δ-good} Then a theorem The capacity of a noisy channel equals the maximum over input distributions of the mutual information between input and output. [Shannon ’49] 2->4 norm Define ||x||p := (𝔼i |xi|p)1/p Let A2Rm£n. ||A||2!4 := max { ||Ax||4 : ||x||2 = 1} How hard is it to estimate this? optimization problem over a degree-4 polynomial SDP relaxation for 2->4 norm where L is a linear map from deg ≤k polys to R L[1] = 1 L[p(x) (𝔼i xi2 – 1)] = 0 if p(x) has degree ≤ k-2 L[p(x)2] ≥ 0 if p(x) has degree ≤ k/2 Converges to correct answer as k∞. [Parrilo ’00, Lasserre ‘01] Runs in time nO(k) Why is this an SDP? Constraint: L[p(x)2] ≥ 0 whenever deg(p) ≤ k/2 p(x) = ∑α pα xα α= (α1 , …, αn) αi ≥ 0 ∑iαi ≤ k/2 L[p(x)2] = ∑α,β L[xα+β] pα pβ ≥0 for all p(x) iff M is positive semi-definite (PSD), where Mα,β = L[xα+β] Why care about 2->4 norm? Unique Games (UG): Given a system of linear equations: xi – xj = aij mod k. Determine whether ≥1-² or ≤² fraction are satisfiable. Small-Set Expansion (SSE): Is the minimum expansion of a set with ≤±n vertices ≥1-² or ≤²? UG ≈ SSE ≤ 2->4 G = normalized adjacency matrix Pλ = largest projector s.t. G ≥ ¸P Theorem: All sets of volume ≤ ± have expansion ≥ 1 - ¸O(1) iff ||Pλ||2->4 ≤ 1/±O(1) quantum states Pure states • A quantum (pure) state is a unit vector v∈Cn • Given states v∈Cm and w∈Cn, their joint state is v⊗w∈Cmn, defined as (v⊗w)i,j = vi wj. • u is entangled iff it cannot be written as u = v⊗w. Density matrices • ρ satisfying ρ≥0, tr[ρ]=1 • extreme points are pure states, i.e. vv*. • can have classical correlation and/or quantum entanglement correlated entangled when is a mixed state entangled? Definition: ρ is separable (i.e. not entangled) if it can be written as ρ = ∑i pi vi vi* ⊗ wi wi* Sep = conv{vv* ⊗ ww*} probability distribution unit vectors Weak membership problem: Given ρ and the promise that ρ∈Sep or ρ is far from Sep, determine which is the case. Optimization: hSep(M) := max { tr[Mρ] : ρ∈Sep } monogamy of entanglement Physics version: ρABC a state on systems ABC AB entanglement and AC entanglement trade off. “proof”: If ρAB is very entangled, then measuring B can reduce the entropy of A, so ρAC cannot be very entangled. Partial trace: ρAB = trC ρABC Works for any basis of C. Interpret as different choices of measurement on C. A hierachy of tests for entanglement Definition: ρAB is k-extendable if there exists an extension with for each i. all quantum states (= 1-extendable) 2-extendable 100-extendable separable = ∞-extendable Algorithms: Can search/optimize over k-extendable states in time nO(k). Question: How close are k-extendable states to separable states? 2->4 norm ≈ hSep A = ∑i ei a iT Easy direction: hSep ≥ 2->4 norm M = 𝔼i ai aiT ⊗ ai aiT ρ=xx* ⊗ xx* Harder direction: 2->4 norm ≥ hSep Given an arbitrary M, can we make it look like 𝔼i aiai* ⊗ aiai* ? Answer: yes, using techniques of [H, Montanaro; 1001.0017] the dream SSE hardness 2->4 hSep algorithms …quasipolynomial (=exp(polylog(n)) upper and lower bounds for unique games progress so far SSE hardness?? 1. Estimating hSep(M) ± 0.1 for n-dimensional M is at least as hard as solving 3-SAT instance of length ≈log2(n). [H.-Montanaro 1001.0017] [Aaronson-Beigi-Drucker-Fefferman-Shor 0804.0802] 2. The Exponential-Time Hypothesis (ETH) implies a lower bound of Ω(nlog(n)) for hSep(M). 3. ∴ lower bound of Ω(nlog(n)) for estimating ||A||2->4 for some family of projectors A. 4. These A might not be P≥λ for any graph G. 5. (Still, first proof of hardness for constant-factor approximation of ||¢||24). positive results about hierarchies: 1. use dual Primal: max L[f(x)] over L such that L is a linear map from deg ≤k polys to R L[1] = 1 L[p(x) (∑i xi2 – 1)] = 0 L[p(x)2] ≥ 0 Dual: min λ such that f(x) + p(x) (𝔼i xi2 – 1) + ∑i qi(x)2 = λ for some polynomials p(x), {qi(x)} s.t. all degrees are ≤ k. Interpretation: “Prove that f(x) is ≤ λ using only the facts that 𝔼i xi2 – 1 = 0 and sum of square (SOS) polynomials are ≥0. Use only terms of degree ≤k. ” “Positivestellensatz” [Stengel ’74] SoS proof example z2≤z $ 0≤z≤1 Axiom: z2 ≤ z Derive: z ≤ 1 1 – z = z – z2 + (1-z)2 ≥ z – z2 (non-negativity of squares) ≥0 (axiom) SoS proof of hypercontractivity Hypercontractive inequality: Let f:{0,1}nR be a polynomial of degree ≤d. Then ||f||4 ≤ 9d/4 ||f||2. equivalently: ||Pd||2->4 ≤ 9d/4 where Pd projects onto deg ≤d polys. Proof: uses induction on n and Cauchy-Schwarz. Only inequality is q(x)2 ≥ 0. Implication: SDP returns answer ≤9d/4 on input Pd. SoS proofs of UG soundness [BBHKSZ ‘12] Result: Degree-8 SoS relaxation refutes UG instances based on long-code and short-code graphs Proof: Rewrite previous soundness proofs as SoS proofs. Ingredients: 1. Cauchy-Schwarz / Hölder 2. Hypercontractive inequality 3. Influence decoding 4. Independent rounding 5. Invariance principle UG Integrality Gap: Feasible SDP solution SoS upper bound Upper bound to actual solutions actual solutions positive results about hierarchies: 2. use q. info [Brandão-Christandl-Yard ’10] [Brandão-H. ‘12] Idea: Monogamy relations for entanglement imply performance bounds on the SoS relaxation. Proof sketch: ρis k-extendable, lives on AB1 … Bk. M can be implemented by measuring Bob, then Alice. (1-LOCC) Let measurement outcomes be X,Y1,…,Yk. Then log(n) ≥ I(X:Y1…Yk) = I(X:Y1) + I(X:Y2|Y1) + … + I(X:Yk|Y1…Yk-1) …algebra… hSep(M) ≤ hk-ext(M) ≤ hSep(M) + c(log(n) / k)1/2 Alternate perspective For i=1,…,k • Measure Bi. • If entropy of A doesn’t change, then A:Bi are ≈product. • If entropy of A decreases, then condition on Bi. the dream: quantum proofs for classical algorithms 1. Information-theory proofs of de Finetti/monogamy, e.g. [Brandão-Christandl-Yard, 1010.1750] [Brandão-H., 1210.6367] hSep(M) ≤ hk-Ext(M) ≤ hSep(M) + (log(n) / k)1/2 ||M|| if M∈1-LOCC 2. M = ∑i aiai* aiai* is ∝ 1-LOCC. 3. Constant-factor approximation in time nO(log(n))? 4. Problem: ||M|| can be ≫ hSep(M). Need multiplicative approximaton or we lose dim factors. 5. Still yields subexponential-time algorithm. SDPs in quantum information 1. Goal: approximate Sep Relaxation: k-extendable + PPT 2. Goal: λmin for Hamiltonian on n qudits Relaxation: L : k-local observables R such that L[X†X] ≥ 0 for all k/2-local X. 3. Goal: entangled value of multiplayer games Relaxation: L : products of ≤k operators R such that L[p†p] ≥ 0 ∀noncommutative poly p of degree ≤ k, and operators on different parties commute. Non-commutative positivstellensatz [Helton-McCullough ‘04] relation between these? tools to analyze? questions We are developing some vocabulary for understanding these hierarchies (SoS proofs, quantum entropy, etc.). Are these the right terms? Are they on the way to the right terms? Unique games, small-set expansion, etc: quasipolynomial hardness and/or algorithms Relation of different SDPs for quantum states. More tools to analyze #2 and #3. Why is this an SDP?